dairy farms for 1965-1991
Empirical application onbookkeeping dairy farms
MattiRyhänen
Departmentof Economics and Management, Universityof Helsinki
ACADEMIC DISSERTATION
To hepresented,with the permissionofthe Faculty
ofAgricultureandForestry ofthe UniversityofHelsinki,forpublic
criticisminAuditoriumXII, UniversityofHelsinki,
Aleksanterinkatu 5, Helsinki, onFebruary3, 1995,at 12noon.
Now that this academic dissertation is completed, the author has the pleasant task ofex- pressing his gratitudetoall persons and institutions that contributedtothe work. The study was made while working as anassistant and acting senior assistant atthe Department of Agricultural Economics, Faculty of Agriculture and Forestry of the University ofHelsinki,
atpresentcalled the Department of Economics and Management.
I wish to express my deepest gratitude to my teacher and supervisor of my research work,Professor Emeritus Viljo Ryynänen, the former Head of the Department of Agricultur- alEconomics, for his expert guidance and friendly supportand encouragement atall stages of my studies. His influencewas significant in arousing my interest in agricultural produc- tion economics and, in particular, in the topic of this study.
I wish tothank Professor MattiYlätalo, teacher of my main subject and second supervi- sor of this study. His contribution was invaluable, especially in the early stages of the empirical study. Also, his support, expertise, and the numerous discussions we had were highly significant in completing this task.
To the Head of the Department of Economics and Management, Professor Karl Johan Weckman, I wish toexpress my sinceregratitude for his support. Iamalso very grateful for the good working atmospherethat, thankstohim,prevails inourDepartment.
Further, I wish toexpress my gratitude toProfessor YrjöVartia, Head of the Department of Economics of the University ofHelsinki, for his supportand personal guidance in utiliz- ing the panel data. I am very grateful to Professor Jouko Sirén, Head of the Agricultural Economics Research Institute, for his advice and the critical and constructive comments, which helpedtoimprove the study. I also wishto thank Ph.D. Mikael Ingberg for inspecting my licentiate’s dissertation and for his suggestions for amplifying it intoadoctoral thesis.
Iwish toexpress my gratitudetothe whole staff ofour Department, and especiallytomy colleagues Lie. Sc. Timo Sipiläinen, M. Sc. Arto Latukka, and M.Sc. Perttu Pyykkönen for their constructive criticism, help, and support. Their fair and encouraging attitude both at work and in free-time will neverbe forgotten.
Mr. Jari Siirilä helped me in recording and preliminary processing of the research data collected from the bookkeeping farms. Together with the Head of the Bureau for Profitabili- ty Studies, M.Sc. Juhani Ikonen and the staff of the Institute, he deserves my deepest gratitude for collecting the data, which took almosta year.I thank M.Sc. Asko Simojoki for finishing the figures included in the study.
The original Finnish text was translated by M.A. JaanaKola, and the final checking of the languagewas done by M.Sc. Terese Forster. I wishto thank both of them fora job well done.
1 have received grants for this study from the August Johannes and Aino Tiura Founda- tion of Agricultural Research, the MTK funds of the Finnish Cultural Foundation, the Academy ofFinland, and the Kyösti Haataja Foundation of Okobank Group, and 1 wish to express my sincere gratitude for the economic support. I also thank the Editorial Board of Agricultural Science in Finland for including my study in their publication series.
Also, I wishtothank my parents, Katri and Vilho Ryhänen, for the practical knowledge and skills I gained when growing and working on a dairy farm in Northern Savo. Practical knowledge has beenan invaluableassetin my research work.
Helsinki,October 1994 Matti Ryhänen
1 Introduction 525
1.1Backgroundfor the study 525
1.2 Objectivesand significanceof the study 526
1.3 Approach to the researchproblem 527
2 Application of the dualapproach of the neoclassical production and costtheoryonFinnish
dairyfarms 528
2.1 Theoreticaldescriptionof theproduction technology in milkproduction 529 2.2 Traditional approach in empirical studyofproduction technology 530 2.3 Applicationof the dual approach inthestudyof the economy ofdairyfarms 531
2.3.1 Economic behaviour 532
2.3.2 Studyof economic activity 532
2.3.3 Costfunction study 534
2.3.4 Properties of the cost function 534
2.4 Comparativestatics 536
2.4.1 Effect ofchanges in input prices 536
2.4.2 Effect ofchanges inoutput 537
2.5 Dualitybetween theproductionand cost function 538
2.6 Definition of elasticities of substitution from the cost function 538
2.6.1 Marginalrateof technical substitution 538
2.6.2 Relationship between theisoquantand isocostcurves 539
2.6.3Elasticities of substitution 540
2.7 Cost flexibility 542
3 Shortruninalong-run study 543
3.1 Staticexamination 543
3.2 Dynamicexamination 544
4 Earlier studies 546
4.1 Finnish studies 546
4.2 Foreignstudies 548
5 Theoretical model for the study 551
5.1 Choiceof the cost function 551
5.2 Theoretical model 553
5.3 Research method 555
6 Agricultural production duringthe research period 556
7 Data and variables 558
8 Results of the study 562
8.1 Descriptivestatistics of the farms included inthe study 562
8.2 Representative dairyfarm approach 563
8.2.1 Shortruninalong-run study 563
8.2.1.1 Autocorrelation functions 564
8.2.1.2 DF-and ADF-tests 564
8.2.2 Empirical models andhypothesistests 566
8.2.2.1 Regressionstatistics and tests 568
8.2.2.2 Empirical models and the theoreticalassumptions 570 8.2.2.3 Empiricalmodels for the post energy crisis period 571
8.2.3 Elasticities of substitution 574
8.2.3.3 Shadowelasticities of substitution 577 8.2.3.4 Summaryof the elasticities of substitution 578
8.2.4Technical change and costflexibility 579
8.3 Dairy farm levelapproach 582
8.3.1 Estimationprocedureforpaneldata set 582
8.3.2Empiricalmodels andhypothesistests 583
8.3.3 Elasticities of substitution 584
8.3.4Technicalchangeand costflexibility 586
9 Examination of the results and conclusions 588
10 Summary 593
References 595
Selostus 599
Appendices
Input substitution and technological development on Finnish dairy farms for 1965-1991
Empirical applicationonbookkeeping dairy farms
MattiRyhänen
Ryhänen, M. 1994.Inputsubstitution and technological developmentonFinn- ish dairy farms for 1965-1991. Empirical application onbookkeeping dairy farms.Agricultural ScienceinFinland3: 519-601. (Department of Economics and Management, P.O.Box27,FIN-00014 University ofHelsinki,Finland.)
The studypresents anattempt togain abetterunderstanding of the inputsubstitu- tion and technological developmentonFinnishdairy farms. The dual approach of the neoclassicalproduction and costtheoryisapplied. Asystem of derived demand and cost functions is estimated using arepresentative dairyfarm data and panel data ofbookkeeping Finnish dairyfarms. The flexible translog cost function is utilized to solve theempiricalresearch problem.The cost functionstudy ischosen, because it makes itpossible tostudy production of farms operating in the area of decreasing average costs.
According totheresults,inputsare for the most part substitutes with each other.
With theexisting production technology, the substitution of inputs for otherinputs is inelastic. Theown priceelasticities arealso inelastic. Technical change is pur- chased feed-saving and other inputs-using. The average annual rate of technical change was 1.3percent. The newproduction chainsresulting from technicalchange have made it possible to increase the size ofdairy farms. Increasing the size of dairyfarms should be allowed so inorder to make itpossible toutilize the advan- tages related to the economies of size.
Key words: dual approach, elasticity of substitution, flexible functional forms, inputdemand,panel data, representative farmdata,technicalchange
1 Introduction 1.1 Background for thestudy
Agricultural production in Finland has undergone arapid change since the early 19605. Changes in agriculture areclosely linked tothe development of the national economy. The rise in the standard of living creatednew jobs, which attracted a la- bourforce,especially from small farms. The pop-
ulation of the countryside has decreased byover
halfa million people during this period of time, although the population of Finland has increased by about 300,000persons.
As aresult of the change, the labour force tied to agriculture has dropped from 450,000 to the present estimation of 150,000 persons (Etla 1993, p. 106). At thesame time,the netcapital stock of agricultural production buildings, machinery, and implements has increasedover 1.5 times. The rise
in ihe total productivity in agriculture has varied between2% and 4%. However, the changes have not led to improved profitability in agriculture (Ylätalo 1987, p. 2, 70-71).
The incomes of agricultural entrepreneurs have remained smaller than those of people working in other sectors (Puurunen 1990, p. 77-88). The low incomes and the fact that farmers are very much tiedto their work has made it difficult to find young people who are willing to continue farming, especially on dairy farms (Ryhänen 1989, p. 50). This has resulted in a rapid de- crease in the number of dairy farms. During the research period close to 200,000 dairy farms stopped production. At present, the number of dairy farms is under 35,000.
Finnish agricultural policy has been character- ized by conflicting interests and widely different views on agricultural policy between interest groups like political parties and labour market organizations. In this kind of atmosphere, there- alization of any long-term agricultural policy has been problematic. In addition, as a result of the vast overproduction, the objectives in the regula- tion of milk production have mainly concerned restricting production. Consequently, dairy farms were largely excluded from the active develop- mentmeasures,for example, size restrictions were placed ondairy farms.
How did the problems comeabout? After the Second World War, itwas not possible toimport food. Instead, the Finnish government encour- agedmoreefficient and increased agricultural pro- ductiontomake up for the food shortages. At the same time,the “social problem” of homeless fam- ilies was addressed. This resulted in growth in the number of farms and a decrease in the aver- age farm size. As aresult of the increased arable landareaand themore efficient production in the 19605, the shortage of food gradually turnedto overproduction.
In addition to the need to solve the present problem of overproduction, the need to develop the dairy farms increased atthe same time. De- spite measures torestrict production, in the long run economic factors will determine the trends in agricultural production. Small, labour intensive
dairy farms will become unprofitable and give up production. The present small average farm size and the relatively large share of small farms make it impossible to use inputs in an efficient way atthe level of the national economy.
When developing dairyfarms,it is essential to know how the resources should be allocated to milk production. There are very few reliable re- search results available on the use of production inputs on dairy farms in Finland. Consequently, the study of theuseof these inputswas chosenas the central objective of this study.
The initial assumption of the study is that a milk produceractsso astooptimize theeconom- ic result of production. In addition, it is assumed that labour and capital are the factors by means of which milk producers organize the production of their farms. As a result, the use of labour and capital also describes indirectly the abilities and entrepreneurship of the milk producer.
In addition to optimizing the economic result, the milk producer may have other objectives, for example, enough leisure time, avoidingrisk, and independence (Brandes et al. 1980). These fac- tors are impossible to measure in an exact and reliable way, so that research on the effects of these factors is excluded from the study.
1.2 Objectives and significance of the study
The objective of the study is to examine the ef- fect of the change in the relative prices of inputs and the technological development of production onthe derived demand for inputs of dairy farms.
In otherwords, the aim isto study how the change in the relative prices of inputs and technological development affect the production technology of a dairy farm. The study provides informationon towhatextentinputs cansubstitute each other, as well as on how technological development has influenced the demand of individual inputs, re- sulting in an increase in the demand for certain inputs and a decrease in the demand for the others.
There has been very little researchonthe pos- sibilities for input substitutiononFinnish dairy farms. However, awareness of these possibilities is important from the viewpoint of the needs of both dairy farms and agricultural policy. Knowl- edgeon the relations between the inputs is essen- tial in developing dairy farms and in directing agricultural production.
Accordingto the production andcost theory,a change in the relative prices of inputs affects the actions of dairy farms. When relative prices change, the milk producer should know which inputs are substitutes and which ones are com- plements of each other, and theextent to which input substitution is possible sothat the producer could strive to adjust production in an optimal way. In the study it is tested if milk producers follow the assumption of rationalbehaviour,which is assumed in the production andcosttheory. It is also examinedto whatextentthe production tech-
nology of the farm has changed, along with the technological development. It is necessarytoknow the earlier technological development in estimat- ing the effect of future changes on the demand for inputs. In addition, in making decisions on agricultural policy, e.g., incidence oftaxation, it is usefultoknow how the change in the price of anindividual input influences the demand.
In mostFinnish studies and publications relat- ed to the use of labour and capital, the use of capital in agriculture has mainly been examined at the macro level (Ihamuotila 1972, 1983, Ylätalo 1987). In the macro level studies the main emphasis has been on finding out the capi- tal stock of agriculture. In addition, the studies have concentrated onresearch into the produc- tivity of agriculture,as well as of the changes in the amountsof capital and labour inputs used. In the study of the reasons for change in the agri- cultural sector,evidence for the theories has usu- ally been searched for in the observation series included in the total statistics.
The studies madesofar have provided impor- tantresults onthe significance of capital in agri- cultural production, but these results give onlya rough picture of theuse of inputs on dairy farms.
There is very little empirical research results avail-
able on the use of labour and capital, and the relations between themonFinnish dairy farms.
1.3 Approach tothe research problem In this study, the neoclassical production and cost theory is applied in examining theuse of inputs on dairy farms. The dual approach is used to solve the research problem. Based on the neo- classical production andcosttheory, it is assumed that the production function describes the techni- cal relationship that transforms inputs into out- put, dairy farms aim atminimizing costs, and the prices of inputs and products areexogenous.
The neoclassical framework is well suited for both theoretical and empirical factor substitution studies, and thus it was chosen as the theoretical point of departure for the study. According to Varian (1992, p. 23), profit maximization has been the basic assumption inmosteconomic anal- yses of firm behaviour. This assumption provides an exact framework for the analysis and testing of the results. The neoclassical production and cost theory describes the production process of an enterprise, which means that conclusions can be made logically onthe basis of the theory.
The study is basedon a long-runexamination, which means that all costs are assumed to be variable costs. In the shortrun all costs of milk production cannot be considered to be variable, but some of thecosts, e.g., thecowshed, is in the short run quasi-fixed. By means of the cointe- grated time series and theerrorcorrection model linking these, it is possible to incorporate along- run and short-run study through the theory of
statistics (Hendry 1991, Engle and Granger 1991). In connection with the attempt to solve the research problem, it will be examined if the input demand system of milk production can be modelled intoan errorcorrection model.
in the beginning of the 19705, after the energy crisis,research in general economics and in agri- cultural economics was to an increasing extent
directed to the investigation of the substitution possibilities among inputs. The study of the sub-
stitution possibilities also gavea powerful impe- tus to the development ofnew functional forms.
In this century economists have strived tofind a connection between the mathematical methods of representation and the neoclassical production function. However, some economists tooka dif- ferent view of the problem (e.g., Diewert 1971, deJanvry 1972, Christensenetal. 1971, 1973).
They started developing flexible functionalforms, which greatly increased the possibilities for an empirical application of the neoclassical produc- tion and cost theory. These functional forms are used tosolve the research problem.
The theory of the study and the grounds for the approach chosen arepresented in Chapter 2.
The first part of the chapter is based on litera- ture, and it deals with the properties of the pro- duction possibility setand the production func- tion by examining these from the viewpoint of the milk producer. Then the dual approach, which
is the main approach in the study, is presented. In the dual approach the main emphasis ison exam- ining the derived demand of inputs and input sub- stitution, as wellasin determining the elasticities of substitution. Chapter 3 provides a theoretical examination of the possibilities toincorporate the short-run and the long-run study. The Finnish and international publications related to the research topicarereviewed in Chapter 4, and the theoreti- cal model of the study is presented in Chapter 5.
The development of Finnish agriculture during thepast three decades is described in Chapter 6, and the research data is presented in Chapter 7.
The results of the study are presented in Chap- ter8. Chapter 9 presents an examination of the results and conclusions, based on the research results and the theory, and on the basis of the information provided by theresults, the possibil- ities todevelop the production of dairy farms are discussed.
2 Application of the dual approach ofthe neoclassical production and cost theory on Finnish dairy farms
The neoclassical production and cost theory, and the basic findings it has provided, have estab- lished their position in economics during thepast decades. No serious alternatives to the neoclassi- cal production and cost theory have been devel- oped. The most remarkable progress in the re- search onproduction economics has been achieved in improving the generality of the results of the neoclassical production and cost theory. In the past few years the main objective in the research of the production and cost theory has been to make it possible to describe more complicated production processes thanearlier,andto improve the testability of the models used in the empirical study.
The dual approach and the flexible functional forms developed in the past two decades have expanded the application possibilities of the pro- duction and cost theory considerably. There are concrete advantages in the dual approach in ap-
plied economics, in particular. The demand and supply functions determined directly from the dual function simplify the analyses considerably, com- paredto the primal studies, because in determin-
ing themno nonlinear equation systems thatare difficult orimpossible tosolve areneeded. In the dual approach it is possible to describe the pro- duction technology in certain circumstances equiv- alently by means of both the primal function and the dual function(Diewert 1982, p. 535-547). In this case,it is assumed that the prices of inputs indicate thesame things of production technolo- gy as theamounts of inputs, and that enterprises aim at maximizing profits and/or minimizing
costs.
It is essential for a milk producertobe famil- iar with the production technology, because this determines the limits within which the actions of the dairy farm are possible. In the beginning of the study, a brief description of the production
technology in milk production is presented. In this study the economic behaviour of the milk producer receives special emphasis, so that an exhaustive account of the dual approach that is central in the study is presented, to the extent that it is needed to solve the empirical research problem,.
2.1 Theoretical description of the production technology in milk production
When planning milk production, the first task of a milk producer is to examine the combinations of inputs by which production is feasible, within the framework of the existing production tech- nology. The technical production possibilities fac- ing dairy farms canbe described bya production possibility set, which gives all feasible input and outputcombinations. Thus the production possi- bility set is a subset of R". Consequently, the production possibility setprovidesacomplete pic- tureof die production possibilities ofadairy farm.
The production plan ofamilk producer canbe presented as a list that includes both inputs and products. Some goods, like barley, on a dairy farm may be, simultaneously, input (feed for ani- mals) and product (cultivation of grains).Thus, it is expedienttopresent the inputs and productsas netputs. If thenetputofagood i is positive (neg- ative), the dairy farm producesmore (less) of the good i than it uses.Thiscan be presented exactly as a production plan on a netput vector yeR", where y, is negative if the good i is a net input, and positive if the good i isapositive netput.
At the loss of generality, the inputs and out- puts aredealt with separately, because this kind of representation is considered intuitively useful (e.g., McFadden 1978
a,
p. 6, Chambers 1988, p. 252, Varian 1992, p. 2-3).Thus, x=(x„...,xn) eR" describes a positive input vector, and y = (yi,...,ym)GR+ describesa positive output vector.With the existing production technology the pro- duction possibility set T, which gives all techni- cally feasible combinations of (x,y) can be de- finedasfollows:
T={(x,y):x canproduce y} (1)
The properties required of T arepresented in Appendix
I.
The technological knowledge and the laws of nature determine largely the production possibil- ity setofa dairy farm. However, the production possibility set may be more limited in practical decision-making on a dairy farm due to,e.g.,re- strictions onproduction, natural conditions, and environmental factors. Also, on a dairy farm the production possibility set in the short run may differ from that ofalongrun,sothat in an inves- tigation of dairy farms it is essential to separate short-run and long-run production plans from each other. In short-run production plans quasi-fixed factors are considered to be factors that restrict production. In the longrun all inputs canbe con- sidered variable inputs. In practice, distinguish- ing the shortrun and the longrun, in milk pro- duction, from each other is not a dichotomous phenomenon.
The short-run studycan be presented exactly as the short-run production possibility set T(z)CT, in which thevectorze
R+
describes the production restrictions in the shortrun. Vector z is,for example, a list of maximumamounts of inputs and products thatarepossible in the short run. Amore detailedaccount of issues related to short-run and long-run studies is presented in Chapter 3.When the production technology in milk pro- duction is examined, the production possibility setpresented abovecanbe simplified. Other prod- uctsrelated tomilk production canbe considered by-products of milk production, in which case we shift from the technically efficient production plans, transformationfunction, to the production function, which describes the maximum scalar output as a function of inputs, instead of maxi- mal vectors of netput. Thus, the model for the production technology on farms that specialize in milk production can be formulatedso that the outputis presented as one aggregate output.
In general, heavily aggregate data have been used in empirical studies basedon the production andcost theory that have been applied toagricul-
ture (e.g.,Ray 1982, Glass and McKillop 1990,
Ryhänen 1992). According to the production and cost theory, as little aggregation as possi- ble is an objective, sothat in morerecent stud- ies the degree of aggregation has been reduced, and the studies mainly concentrateon examining the production technology ofone production line (e.g., Tiffin 1991, Thijssen 1992
a,
b). In these studies thenetput bundle (x,y) has been formed from the scalar y and vector x, so that x can produce y.From the viewpoint of this study, it is expedi- ent to present the production technology as an input requirement set, which canbe presented as follows:
V(y)=(xeRJ: (y,x)eT} ={x: f(x) >y}. (2)
According tothe definition given above, V(y) is theset of positive inputvectors x =(x,,...,x„), which produce at least output y. The input re- quirement setcorresponds to the traditional iso- quant, except that it also includes non-efficient input bundles. The boundary of V(y) atacertain output level is the same as the isoquant of this outputlevel(seeUzawa 1964, p.216). Thus, the isoquant is thesetof efficient inputvectors, which produce exactlyoutputy.
In the traditional approach of the neoclassical production and cost theory, production technolo- gy ispresented bymeansof the production func- tion indicating the physical and technical rela- tions (e.g., Heady 1952, Bradford and Johnson
1953). Production function describes the techni- cal relation between the inputs and output, which meansthat it does notinclude any economiccon- tent.The production function is presented asfol- lows:
y=f(x). (3)
where the scalar y is the maximum output that can be produced by means of the existing pro- duction technology in a certain period of time.
Positive output can be achieved by utilizing the non-negative input vector x = (x,,...,xn) in pro- duction.
Independent of how the milk producer defines the supply of his product, it is profitable for him toproduce this output atas low a cost aspossi- ble. Fromaneconomic viewpoint, the production possibilitysetand the production function deter- mine the limit to the optimization problem. The production function defines unambiguously the technical restriction of the optimization problem.
This study is basedon the assumption that there is aproduction function for dairy farms, which describes the connection between the inputs used during acertain period of time and the maximum outputproduced bymeansof these inputs. When analysed from the economic viewpoint, the pro- duction function is expectedtohave certain prop- erties, which meansthat the production function mustbe theoretically well defined (see Appendix
I and 2).
2.2 Traditional approach in empirical study of production technology The properties of the production function pre- sented in Appendix 2 have usually been adequate for the purposes of theoretical analyses of the traditionalorprimal approach, but in mostcases they have not been fully adequate for empirical analyses (Chambers 1988, p. 36). In empirical studies it has often been necessary to setaddi- tional restrictions onthe production functions in ordertomake the production andcosttheory and the research techniques consistent with each oth- er. In connection with this, assumptions on the homogeneity, homotheticity, and separability have been made.
In empirical studies homogeneous production functions of degree k have frequently beenused, in which f(ax) = akf(x). The best known homo- geneous production functions are the Leontief, Cobb-Douglas and CES production functions (Nadiri 1982, p. 457-459). Homogeneous pro- duction functions have proven useful in empiri- cal applications. For example, the proportional changes in all individual inputs that change the scale of production arereflected exactly by the
same proportional change in an aggregate in- put.Agricultural economists have frequently used the Cobb-Douglas production function in their analyses.
The production function is definedashomoth- etic, if it can be presented in the form f(x) = g[h(x)], where h(*) is homogeneous of degree one, g(») is a monotonic function, and g and h are twice differentiable (Varian 1992, p. 18). Ho- mothetic production functionsarefunctions which generate linear expansion paths emanating from the origin, when the prices of inputsareconstant.
Every homogeneous production function is also homothetic, but homothetic production functions also include non-homogeneous production func- tions,in which thereturns to scale may vary as a function ofoutput(Sandler and Swimmer 1978, p.357).
When the number of inputs increases, the esti- mation ofan empirical model has usually been considereddifficult, evenimpossible. This restric- tion has often forced economists to employ a smaller number of inputs in empirical analyses than in theoretical analyses. Homogeneous tech- niques and a relatively small number of inputs has often proven tobe auseful approach in solv- ing empirical problems.
As a definition, inputs X; and Xj are separable from the input xk (Blackorby et al. 1977, p. 197),if
a a//aV
dxt
[df/dxjj
(4)i.e., the slope of the isoquant in the dimension i,j is not affected by what occurs in dimension k.
The location of the isoquant in the dimension i,j may change as a result ofachange in the dimen- sion k.
The assumptions of the homogeneity, homo- theticity, and separability of the production func- tion restrict the possibilities to use the different functional forms in empirical analyses considera- bly. In general, this has not been considered a
major problem. Econometric techniques have been applied, according to the restrictions mentioned above. The assumption of strict separability and
homogeneity (Cobb-Douglas, CES) provided a good point of departure for the estimation ofeco- nomic parameters for many decades. However, compromises hadtobe made between the gener- ality, and the analyticalorempirical flexibility.
2.3 Applicationof the dual approach in the study of the economyof dairy farms
In the research on agricultural economics based on the production and cost theory practiced in Finland, the physical and technological produc- tion possibilities of agriculture have traditionally been examined by meansof the production func- tion (e.g.,Ryynänen 1970, Hemilä 1983, Yläta-
lo 1987). The dual approach was chosen for this study because it makes it possible to formulate the model applying the production and cost theo- ry directly into aform that describes the causal economic relations. Also, in an empirical study the research methods of the dual approachcanbe more easily and exactly dealt with mathematical- ly and in the data processing than the methods of the traditional approach, which meansthat there are significant advantages in the use of the dual functions (profit and cost function), compared to the traditional approach.
The development of the dual approach of the production and cost theorywas started by Hotel- ling and Samuelson in the
1930 s and 1940 s
(Mc-Fadden 1978
a,
p. 5). According to McFadden, the publication of Shephard in 1953, where the duality between thecost and production function is presented, canbe considered the breakthrough of the dual approach. In the 19605, the dual ap- proach was further developed (e.g., McFadden1963, Uzawa 1964). From the viewpoint ofem- pirical studies, the development of flexible func- tional formswas decisive (Diewert 1971, Chris-
tensenetal. 1971, 1973).In the 19705, McFad- den developed, with his colleaguesatthe Univer- sity of Berkeley, the foundations for the applica- tion of the dual approach in empirical study. Ac- cordingtoChambers(1988,p. 121), the research
results of McFadden and his colleagues caused the research in empirical production economics to turn in completely new directions.
2.3.1 Economic behaviour
Accordingto the neoclassical production andcost theory, the basic assumption in this study is that entrepreneurs aim atmaximizing profit. Accord- ing to the theory, the actions ofan entrepreneur are optimal, when the marginal revenue is equal to the marginal cost of production. This deter- mines theoutput level chosen for production and the quantity of inputs used by an optimally act- ing entrepreneur. When an entrepreneur deter- mines his optimal activity, he also has to take into account the factors restricting production.
The technical production possibilities and themar- kets (the consumption of the products) are the most common factors restricting production in the enterprises.
The behaviour of the milk producer in themar- kets can be approximated by means of the as- sumption of pure competition (see e.g., Debertin
1986, p. 9-11). In the conditions of pure compe- tition theentrepreneur is assumedtopossesscom- plete knowledge of the prices of both products and inputs.
From the viewpoint of a milk producer, the input markets in milk production may be consid- ered tofulfill the preconditions for pure competi- tion, at least withrespectto the fact that it is not possible for them to influence very much the prices of inputs they have acquired through their ownactions. The product markets ofadairy farm arenot basedonpure competition, rather the prices of products are agreed on in advance during the farm income negotiations. At the farm level the quantity of milk production is regulated by the state, which means that the equilibrium of milk production canbe described by means of profit maximizationatacertain level ofoutput, instead of unrestricted profit maximization.
If the conditions mentioned above prevail, milk producers cannot influence the prices of inputs or products in a decisive way through their own actions. Thus,from the viewpoint ofa milk pro-
ducer, the prices of both inputs and products are exogenous.
2.3.2 Study of economic activity
On the basis of the previous chapter it can be assumed that amilk producer takes the prices of both products and inputs as given. The aggrega- tion assumption made in connection with the the- oretical description of the production technology in milk production makes it possible to write the problem of profit maximizationon a dairy farm in the following form:
H(p,w)=Max (pf(x)-wx) =Max (py-C(w,y)), (5)
x>o y>0
in which the scalar p is the price ofoutput, vec- torw =(W|,...,wn ) presents the prices of inputs, and the quantity of inputs used is described by the vector x =(x,,...xn). The profit function ofa dairy farm Tl(p,w) gives the maximum profit as the function of prices. Applied to the research problem, the restricted profit function canbe pre- sented equivalently with thecostfunction C(w,y)=Min {wx: xeV(y)), (6)
x>o
where the cost function indicates the minimum cost atacertain output level, when the prices of inputs are equal to w. When the level of output has been determined in advance, the return is also fixed, in which case the profit of a dairy farmcanbe maximized by minimizingcosts.The connection between the ways of presenting the equation (5) is presented in Appendix 3.
Thecost function is more useful than the prof- it function in the study of the economic actions of Finnish dairyfarms, because in thecase of the costfunction both decreasing and increasing av- eragecosts arepossible. The profit function can be used in the study of the economic actions ofa dairy farm only in the area of the average cost curve where the average costs increase.
The use of the profit function in the study of the economic actions of dairy farms canbe illus- tratedas follows. Let us assume that conditions of pure competition prevail, and the behaviour of
the milk producer is in accordance with the profit maximization assumption. In this case, the mar- ginal cost is equal to the marginal revenue. In these conditions the marginal revenue isconstant, and is equaltothe price of product p, sothat the elasticity of sizecan be calculatedasfollows:
(7) AC/p=C(w,y)/(py),
where AC is the average cost.In formula (7) it can be seen directly that a rationally behaving milk producer produces milk only if the ratio according to formula (7) in the longrun (all in- puts arevariable) is not larger thanone,because profit cannot be negative in the long run (cf.
Chambers 1988, p. 124-125). If the ratio is more than one, arational milk producer should expand the enterprise or give up production gradually.
Support for this reasoning can be found in the current situation in milk production in Finland.
Within the framework of the technology it would be possible toincrease milk production on indi- vidual dairy farms, but the milk quotas prevent the increase of the farm size.
The development of dairy farms seems con- sistent with the theory. Since anincrease in farm size has been prevented, many farms have given up milk production. During 1985-1993, over 30,000 dairy farms quited production. At the end of 1993, the number of dairy farms was a little over34,000. The rapid decrease in the number of dairy farms provides evidence for the fact that the preconditions for profitable milk production have been weak. On the basis of this, it seems that dairy farms operate, on the average, in the area of decreasing average costs, for which the profit function hasnot been defined. In this case the restricted profit function, which canbe pre- sented equivalently with the cost function, can be usedto solve the research problem.
In the literature on agricultural economics the economies of size isaproblemareathat has been dealt with extensively. Accordingto the literature the research results deviate from each other, but most studies have arrived at the conclusion that in agriculture the average costsdecrease as afunc- tion of output (see e.g.. Heady 1952, p. 349-
350, Hoch 1976, p. 746-748, Smith etal. 1986, p. 719-720, Castle 1989, p. 574-577). It should be noted, however, that the empirical research results related tothe economies of size are local, which means that they do not tell the absolute truth.
According to Quiggin (1991, p. 36), in the world of risk and uncertainty the optimumoutput level will be either in theconstant or increasing returns to scale area. This is characteristic in ag- ricultural production, and most studies on the economies of size of agricultural enterprises have provided evidence for this (cf. Heady 1952, p. 350). In addition to the traditional U-shaped average cost curve,evidence for anL-shaped av- erage cost curve has also been found, in which case the production technology is first described by the increasing returns to size and finally by the constant returns to size.
Even if, accordingto the theory, the costfunc- tion study isa moreuseful approach in theexam- ination of the economic behaviour of Finnish dairy farms than the profit function study, there are also limitations in its use. In the cost function study it is assumed that changes in the prices of inputs do not affect the level of output, so that the indirect effect of the changes in the prices of inputs on the output level is not taken into ac- count.
The profit function study makes it possible to define the change in endogenous output. Milk production restrictions began in 1970, and the measures torestrict production finally ledto the production quota system in 1985. In the produc- tion quota system it is not profitable for a dairy farm toexceed its milk quota,which meansthat the theoretical basis for the definition of the change in endogenous outputdoes not exist. The measures torestrict milk production have been presented in detail,e.g., in Kola’s (1991, p. 122-
126) study onthe regulation of production.
The cost function study is appropriate for the examination of the behaviour of Finnish dairy farms, because it is well suited for the study of the behaviour of enterprises that do not operate in the conditions of pure competition in their prod- uct markets (Jorgenson 1986, p. 1884).In regu-
lated production the price ofa product has been determined, e.g.,by agreements, as is thecase in milk production. Thus, the demand for milk is determined according toaregulated price, which means that, under these preconditions, the level ofoutput is exogenous. In this case, the neces- sary conditions for the optimum in milk pro- duction canbe derived from the cost minimiza- tion.
It should be noted that thecost function study is notincompatible with profit maximization be- haviour. The profit functioncanbe presented, so that at a certain output level, the difference be- tween the total revenue and total costs is maxi- mized. In other words, when the profit is maxi- mized ata certain output level on a dairy farm, thecostsare, atthe same time, minimizedatthis outputlevel. If this was notthe case, it would be possible to produce milk cheaper, which would no longer bea question of profit maximization.
This canalso be presented by means of elas- ticities. Let us assumethat the maximum profit is achieved atthe output level y*. In this case the elasticity of size solved from the profit function is equal tothe elasticity of size solved from the costfunction atthesame output level,i.e. AC(y*)/
MC(y*).This connection is realized since the profit maximizing first order condition states that the output price is equal tothe marginal cost at this point (formula 7).
2.3.3 Cost function study
In the latter part of this study the economic be- haviour ofadairy farm is examined onthe basis of the cost function. In the study it is assumed that a milk producer chooses the quantity of in- puts to be used, so that the costs are minimized
atthe output level determined in advance. In the form of a cost function the research problem is presented asfollows:
c =C(w,y)=Min {wx:xeV(y)), (8)
x£0
where w =(w,,...,wn)is thevectorof the positive input prices. A milk producer chooses for pro- duction the input bundle x, which minimizes the
N
cost c = wx = ZwjX| ata given level ofoutput.
i=I
Thus, thecostfunction defines the minimumcost of production, whenoutputy, which is determined in advance, is produced during a certain period of timeatinput prices w. Consequently, in theory and in practice minimizing costs is equaltomax- imizing profitatacertain level ofoutput.
2.3.4 Properties of the costfunction
In ordertoachieve economiccontent,restrictions must be made on the cost function, as was the casewith the production function. C(w,y) depends on the production technology, which means that the production technology determines the limits within which the minimization ofcosts is possi- ble (see Appendix 2). The properties of the cost function according tothe technology restrictions are(Chambers 1988, p. 52-59):
1.
C(w,y) >O, w,y>O, non-negativity;2. if w’> w,then C(w’,y)>C(w,y);
3. concaveand continuous in w;
4. C(aw,y)=aC(w,y), a>0;
5. if y>y’, then C(w,y)> C(w,y’);
6. C(w,o)=0,no fixedcosts;and
7. 3C(w,y)/9w|= X|(w,y),X| is the demandvector of inputs. C(w,y) is twice continuously differ- entiable.
According toproperty 1 of the costfunction, it is impossible to produce apositive output with- out costs. Accordingtoproperty 2, the rise in the price of any inputx, cannotreducecosts.
Property 3 is difficultto perceive intuitively.
Let us assume that the price ofone input chang- es, while the prices of other inputs remain un- changed. If the price ofan inputrises, the costs cannot decrease (property 2), but they may in- crease at a decreasing rate. This is possible be- cause, as the input becomes more expensive and the prices of other inputs remain the same, the cost minimizing dairy farm substitutes other in- puts for the relatively more expensive input. Ac- cording toproperty 3, inputs are substitutes with each other, if the dairy farm can shift from the relatively expensive inputs to the use of other
inputs. In this case, the increase in the price of input Xjreduces its use, and increases theuse of other inputs. If C(w,y) is linear, the input bundle is completely fixed, which means that substitu- tion between inputs does notoccur.
Property 4 describes positive linear homoge- neity, where only relative changes in the prices have an effecton the optimum. If the prices of inputs change in thesameproportion, the choice of inputs that minimizes costs does notchange.
According to property 5, increasing output can- not result in lower costs. According toproperty 6, a zero outputdoes notcause anycosts, sothat the examination is directed to the long-run cost function.
According to 7, the twice differentiable cost function has a property that is called the Shep- hard’s Lemma. Bymeans of the Shephard’s Lem- ma it is possible to determine the cost minimiz- ing derived demand functions from the costfunc- tion. The demand for inputs x*, which minimizes thecosts, is equalto the partial derivatives of the costfunction withrespect to input prices W;.Due to the Shephard’s Lemma it is not necessary to definea production function correspondingto the cost function, which means that it is not neces- sary to solve the complex algebra involved in deriving the input demand functions using the production function and Lagrangian techniques (Diewert 1987, p. 692).
According to Binswanger (1974, p. 377), in applied production analysis thereare certain ad- vantagesin thecostfunction study, compared with the production function study:
1.
In a costfunction study it is not necessary to impose the homogenous production function of degree one. The cost function is homoge- nous of degree one in input prices regardless of the homogeneity properties of the produc- tion function, because doubling the prices of all inputs also doubles thecosts, but doesnot affect the input ratios.2. The prices of inputs canbe considered more independent variables than the quantities of in- puts. An entrepreneur makes decisions on the use of inputs on the basis of the exogenous
prices, which means that the quantities of in- putsused become endogenous variables.
3. Inversion of the matrix is notneeded in solv- ing the elasticities of substitution, and thus estimationerrors canbe avoided.
4. The cost function study reduces estimation problems and it is well suited for a translog-
function,because it is linear in logarithms.
Pope(1982) discussed in his article the signif- icance and applicability of the dual approach in the applied economic study. According to him, the approach used in the study should be chosen basedonthe objectives of the study. When defin- ing the elasticities of substitution the cost func- tion is more useful than the production function (Pope 1982, p. 347). The estimation of marginal products from thecostfunction involves thesame problems as the estimation of the elasticities of substitution from the production function.
The cost function study also makes it possible to solve the cost minimizing demandvectoras a multi-stage cost minimization problem, if the number of inputs analysed is great. In this case, it is assumed that production technology is ho- mothetically separable(Fuss 1977, p. 89).
For example, ina two-stagecostminimization problem, the aggregate price index of different types offeed, for example, is first optimized by means of the translog-function. Hulten (1973) has shown that the Divisia index is the best choice among the index numbers. Fuss (1977,p.96-97) has shown that the Divisia index, which is an idealindex, can be described exactlyas a linear- ly homogenous translog-function, so that the ag- gregate prices of the first stage canbe used in optimization in the second stage.
Inanempirical study the modelling of the mul- ti-stagecost minimization problem for milk pro- duction in Finland is problematic, because accu- rate data on, e.g., thecost shares of the different types offeed, are not available. The multi-stage method has been used in the empirical studies of the demand behaviour ofconsumers. It is suita- ble for determining the elasticities of demand and substitution of the submodelsof beverages, food, clothes, etc., after which the elasticities of de-
prices, and twice continuously differentiable with respect to input prices at the point (w*,y*), the Hessian matrix must be a negative semidefinite matrix. Consequently, the derived demand has the following properties (McFadden 1978
a,
p.47-48):
mand and substitution of aggregate consumer goodscanbe examined (see Laurila 1994).
2.4 Comparative statics
2.4.1 Effect of changes in input prices
3x|(w,y)/3wi<O (10)
According to the properties of thecost function, the relative increase in the price ofan inputre- duces itsuse. Letus assume that the prices of all inputs increase simultaneously, so that the price vectorof inputs is multiplied by the positive sca- lar. Thus,according tothe linear homogeneity of the cost function, the costsrise in the same pro- portion as prices. The linear homogeneity of the costfunction (k = 1)meansthat the derived de- mand functions are homogeneous of degreezero in input prices, because the partial derivatives of afunction homogeneous of degree k are homo- geneous of degree k-1 (Chiang 1984, p. 411- 413),i.e., thecost minimizing demand for inputs does not change, because an equal proportionate change in the prices of inputs doesnot leadto a change in the demand for inputs.
3x
i
(w,y)/3wj=öxjCw^yVäWj (H) Property (10) is arestatement ofproperty 2 of the cost function, where arise in the price ofan input cannot increase its demand. Property (11) is a symmetry property, which can be used in testing the properties of thecostfunction, as well as for reducing the number ofparameters tobe estimated (Fuss 1987, p.996-997). Property (11) is a technical result from the differentiability as- sumption of the cost function and the derived demandfunctions, and its use in empirical study may reduceproblems relatedto statistical mathe- matics (Fussetal. 1978,p. 229).The elasticities of substitutionareusually con- sidered a suitable way of measuring the substi- tutability of inputs with each other, because they areunit free measures.The derived demand elas- ticity is regarded as a natural measurement for the substitution between inputs. Using the terms of the costfunction, the derived demand elastici-
ty is presentedas follows:
Derived demandcanbe derived directly from the cost function. Let us assume that C(w,y) is twice continuously differentiableat(w*,y*). Ap- plying Shephard’sLemma, the assumption of dif- ferentiability ensures that thecostminimizing in- put demand function Xj(w,y) exists,and it is once continuously differentiable at (w*,y*) (Diewert 1987, p. 692). Define 5x
i
(w*,y*)/5wj to be theNxN matrix of the partial derivatives of the N derived demand functions withrespect to N prices Wj (i,j = L---.N). According to the Shephard’sLemma,
eg=
=9ln Xi(w,y)/3inWj. (12)
According to the Euler’s Theorem and equa- tion (12), witha linearly homogenouscost func- tion(Fuss etal. 1978,p.232)
3xi(w*,y')/3wj=02C(w*,y*)/3w
i
3wj (9) N2e#
=O (13)where 92C(w*,y*)/3w
i
3is the Hessian matrix of second order partial derivatives of thecostfunc- tion at (w*,y*). According tothe Young’s Theo- rem, the twice continuously differentiable cost function has a property according to which 32C(w*,y*)/3w|3wj is a symmetric NxN matrix.Since the cost function is concave in input
and according toequations (10) and (12)
e„<0. (14)
These results are important from the viewpoint of the empirical study, because they make it pos-
sibleto examine the derived demand behaviour ina systematic way on the basis of the properties of the costfunction. The elasticities of substitu-
tion derived from thecost function are presented in Chapter 2.6.3.
2.4.2 Effect of changes inoutput
Accordingto Hanoch(1975, p. 492), returns to scale canbe presented intwodifferent ways:
1) Returns toscale is the relative increase in out- put, as all input quantities are increased pro- portionally along the scale line from the ori- gin in input space (measured by meansof the elasticity of scale).
2) Returns to scale (size) is the increase in out- put relative to costs for variations along the expansion path in input space, which means thatreturns to size are determined from the expansion path of the enterprise. In this case, the prices of inputs are fixed, and costs are minimized atall output levels (measured by meansof the elasticity of size).
The elasticity of scale and size are equivalent everywhere when the cost minimizing points are located along the scale line (Stefanou and Mad-
den 1988, p. 126). This is realized only for ho- mothetic production functions. Usually the elas- ticity of scale and size deviate from each other.
The difference is presented in Figure
1.
Let us assume that a dairy farm operates at pointA, where the elasticity of scale and sizeare equivalent. In addition, let us assume that the output increases, whereas the prices of inputsre- main unchanged, in which case the dairy farm shifts to point C on a new isoquant. The shift results from the change in theoutput onthe scale line (A->B), and the reallocation of thecost min- imizing input bundle (B->C). The elasticity of size measures the change ofcostsfrom point A topoint C. The elasticity of scale measures the change onthe scale line when weshift from one isoquant to another (A->B). According to this, the measurements are equivalent only if thecost minimizing points are on the scale line. In that
case, the elasticity of size is dependent only on output(seeproof in Sandler and Swimmer 1978, p.354-355).
In milk production the inputratios, presented by the minimumcostcombination of inputs,usu- ally variesas afunction ofoutput.An example of this is the different combinations of the labour and capital inputs onsmall and large dairy farms.
Thus, the elasticity of size is more appropriate in examining the economy of dairy farms than is the elasticity ofscale,because the latter doesnot usually correspondtothe economically best choice (cf. Stanton 1978, p. 729-730).
However, homothetic techniques have been popular in applied production analysis. First, be- cause homotheticity implies that all inputs be- have so that they do not reduce output; and, secondly, in homothetic production functions the optimum input ratios areindependent of the level ofoutput. In addition,the assumption of the homo- thecity of the production function has made it possible to treat the inputs as one aggregate in- put.When thecostfunction is consistent with the homothetic production function, the elasticity of size is independent of the prices of inputs (e.g., Ohta 1974, p.63-65).
Usually thereturns to scale definitionsareglo- bal in nature. However, it is possible that in- creasingreturns to scale (size) prevail in thecase of certain quantities of inputs used,and decreas- Fig. I.Elasticityof scale and size (Chambers 1988,p. 73).
